N2015 P2 Q1

As a tree grows, the rate of increase of its height, \(h\) m, with respect to time, \(t\) years after planting, is modelled by the differential equation

\[\frac{\mathrm{d}h}{\mathrm{d}t} = \frac{1}{10}\sqrt{\left(16 - \frac{1}{2}h\right)}\]

The tree is planted as a seedling of negligible height, so that \(h=0\) when \(t=0\).

1. State the maximum height of the tree, according to this model.[1]
2. Find an expression for \(t\) in terms of \(h\), and hence find the time the tree takes to reach half its maximum height.[5]